Polynomial modified Gaussian

For a distribution expanded around the equilibrium position, the first derivative is zero, and may be omitted, while the second derivatives are redundant as they merely modify the harmonic distribution. Since P0(u) is a Gaussian distribution, Eq. (2.28) can be simplified by use of the Tchebycheff-Hermite polynomials, often referred to simply as Hermite polynomials,3. , related to the derivatives of the three-dimensional Gaussian probability distribution by... 

Naturally, the effects of size and microstrains are expressed as convolution products of the pure profile and the profiles resulting from each of these effects. Therefore, the modified expression of the Caglioti polynomial will depend on the hypotheses made on the shape of the profiles of each of the contributions. If we assume that the size effect leads to a Lorentzian profile, and microstrains to a Gaussian profile, we can write the following expressions ... 

The Gaussian mode is a specific case of the more generalized Hermite-Gaussian (HG) modes these are also referred to as transverse electromagnetic (TEM) modes. The TEM modes carry indices / and m, namely TEM/ where / is the number of intensity minima in the direction of the electric field oscillation, and m is the number of minima in the direction of the magnetic field oscillation (basically, the formula describing the TEMqo mode distribution (Equation (3.4)) is modified by multiplication with so-called Hermite polynomials Him x,y, 2 L). 

Equations (1-23) to (2-28) show that the Gaussian term A is the same for all derivatives, only the polynomial is modified by differentiation. While the Gaussian distribution function satisfactorily describes the shape of electronic spectra in the UV and visible range [28-30], the Lorentzian distribution function is better qualified for spectra such as IR [31] and Raman [32], etc. For analytical bands in the NIR (near infrared) range, a mixture of the Gaussian and Lorentzian function is recommended [14]. 

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